Figure B.12: Probability Weighting Functions
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
p
w(p)
— Juniors —Seniors dashed curves are 95% confidence bands based on percentile bootstrap;
probabilistic risk neutrality holds if w(p) = p; dotted lines mark 0.5
Table B.8: Estimation Results - CRDI-Model
Juniors Seniors
p.e. q0.025 q0.975 p.e. q0.025 q0.975
η 0.345 0.130 0.894 0.235 0.173 0.742
γ 0.795 0.536 0.963 0.719 0.505 0.940
ρ 0.237 0.172 0.300 0.188 0.111 0.251
α 0.497 0.419 0.575 0.614 0.556 0.670
β 0.007 -0.065 0.059 0.098 -0.013 0.189
ln L -7206 -6832
BIC 15247 14438
Subjects 57 53
Observations 1117 1060
Parameters 119 111
95% confidence intervals are derived by the percentile bootstrap method with 9999 replications. The resampling procedure accounts for the panel structure of the data (clusters). Parameters include additional estimates for ˆξifor domain- and individual-specific error variances.
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Appendix C
Preferences or Constraints?
A Rational Explanation for Probability-Dependent Risk Attitudes
This chapter has not yet been published elsewhere.
C.1 Introduction
A large body of empirical evidence, accumulated over the last half century, shows that despite its normative appeal expected utility theory is not suited well as a descriptive theory of choice under risk. None of its key premises has been challenged more severely than the postulate that preferences are linear in probabilities.1 Empirical evidence sug-gests that humans and animals alike systematically violate this assumption (Allais (1953), Kahneman and Tversky (1979), Battalio et al. (1985), Kagel et al. (1990), among others).
They tend to overweight small-probability outcomes and underweight large-probability outcomes. This finding compromises expected utility theory at its core. It appears that risk taking behavior can be characterized only inadequately by the curvature of the utility function. Instead, there seem to be other important factors shaping behavior.
Richer models of choice under risk have been proposed, the most popular class being rank-dependent utility (RDU) models (Quiggin, 1982; Luce and Fishburn, 1991; Tver-sky and Kahneman, 1992).2 RDU models preserve many of the normatively appealing properties of expected utility, but are less restrictive about how preferences depend on probabilities.3 As nonlinear weighting of probabilities seems to be a pervasive feature of individual risk taking behavior (see for instance Bruhin et al. (2010)), RDU models turn out to be a useful descriptive generalization of expected utility.
The availability of better descriptive theories of choice under risk, however, is only one side of the coin.4 For most applications it is inevitable to understand what actually drives risk taking behavior. The design of appropriate incentive schemes, for example, often requires the policy maker to know the reasons underlying behavior. In particular, if the policy maker aims at preventing individuals from taking excessive risks in a particular situation, she must know whether such behavior is due to errors, trait or rational rea-sons.5 Only with this knowledge is it possible to implement policies which affect the right people in the right way. This is of particular relevance for policies obeying the principle of asymmetric paternalism (Camerer et al., 2003). This principle says that corrective interventions geared towards particular behaviors should only have an impact on those
1The independence axiom states that if the decision maker prefers (�) a lottery A over a lottery B, i.e. A � B, then pA + (1 − p)C � pB + (1 − p)C for all lotteries C and all probabilities p ∈ (0, 1) must hold. The axiom restricts the functional form of the model to be linear in probabilities (see Machina (1982) for a more elaborate discussion on this).
2See Gilboa (1987) and Schmeidler (1989) for similar theories of choice under uncertainty.
3Quiggin (1993) provides an extensive discussion on this topic.
4See also Starmer (2000).
5A more specific and contemporary example are incentive schemes which aim at preventing fund managers from investing in too risky assets.
economic agents who behave in an irrational way.6 If policy interventions affect rational agents or are based on incorrect assumptions about preferences, they may be a pure waste of money, or even worse, have adverse welfare effects.
The objective of this paper is to demonstrate that apparent probability distortions can be the result of rational choice. We argue that the carriers of risk taking behavior are not only individual preferences, but also restrictions imposed by the environment, and present the following main insights.
First, it is easy to find situations in which the environment can be identified as a predominant driver of risk taking behavior. We give three examples that underscore the wide economic relevance of such situations and their prevalence in various choice domains.
Without needing ancillary assumptions about the utility function, these examples illus-trate how environmental factors can force decision makers to undertake excessive risks.
Example 1 (Foraging) Imagine a bird in an environment which does not provide much food (e.g. a cold winter). The bird only has very little energy reserves left and, hence, needs to find some food to survive the upcoming night. In its struggle for survival, it is confronted with two foraging alternatives: A) either picking the few remaining grains from a feeding place, or B) picking a larger quantity of grain from the ground where a cat is lurking. If the few grains in the feeding place preclude reaching the minimal subsistence level, i.e. if choosing option A leads to certain death, it must opt for the risky option B, although there is some likelihood of getting eaten by the cat. Only the risky choice involves the possibility of survival.
Example 2 (Consumption) Consider a liquidity-constrained consumer, i.e. a con-sumer who only holds few liquid assets and is not permitted to borrow money. The consumer has to meet her existential needs (e.g. to consume food), has to honor her contracts (e.g. to pay rent) and has to settle her bills. Assume that she has the choice between two investments: A) a certain one leading to a small interest payment (bank), and B) a risky one leading either to a large gain or nothing (stock). For simplicity, assume that the outcomes of these investments are paid out immediately after making the choice.
If the outcome from option A together with the consumer’s liquid assets does not allow her to satisfy her needs, she should opt for option B. This option gives her the chance of escaping from getting prosecuted or starving.
Example 3 (Firm) Assume a firm is running out of liquidity. The management must realize sufficient profits in order to avoid Chapter 11 bankruptcy. It has the choice to
6This excludes the case where rational behavior has negative external effects.
invest in one of two projects: A) a project generating only small profits, but for sure, and B) a risky project generating large profits if it succeeds, but none if it fails. If paying off sufficient debts to avoid bankruptcy or significant costs of distress is not possible with the profits generated by project A, the management should better opt for project B. Only this project allows the company to avert imminent bankruptcy.7
All these examples have a number of features in common. The decision maker must achieve a certain, possibly exogenously given, threshold (minimal energy level required to survive or sufficient profit to avoid bankruptcy). We refer to it as the minimal sub-sistence level. Not reaching it leads to direct negative consequences (death, prosecution or bankruptcy), so-called costs of distress. The commodity the decision maker requires (food or liquidity), however, is scarce in that she only possesses a small quantity of it and only has limited (or no) market access. This latter restriction prevents her from trading another commodity (e.g. money) in exchange to additional units of the commodity of interest, and from intertemporally reallocating units of the commodity in order to have more of it at the disposal today. Therefore, to reach the minimal subsistence level, the decision maker has no other option but choosing the more risky option B. Only with this choice, she may attain a terminal outcome, i.e. the assets at her disposal plus the outcome from the option she has chosen, lying above the required level. These examples there-fore illustrate that the pressure imposed by the environment can have decisive impact on behavior.
Second, we derive a simple model capturing this intuition. We show that risk taking behavior can be adequately modeled using standard expected utility, but taking into account the negative consequences materializing when the minimal subsistence level is not achieved. Being exposed to constraints may lead even rational expected utility maximizers to reveal risk attitudes which naturally depend on probabilities. It is their response to the environment which generates systematic departures from standard behavior. Our model produces a wide range of predictions beyond probability-dependent risk attitudes which dovetail nicely with observed risk taking behavior. Among others, these concern common ratio violations, stake effects, the heterogeneity in risk taking behavior and choice domain dependent risk attitudes. Probably most interestingly, most of the patterns we predict are usually explicitly listed as evidence against expected utility theory. Put differently, our approach constructs the theoretical bridge between the standard preference model
7Consistent with this example, Bowman (1980, 1982) finds that managers of troubled firms take risks that they would not take in other situations. Similar findings are reported for farmers (Kunreuther et al., 1979).
and effectively observed behavior.
Third, our findings have material implications. We already mentioned that the design of appropriate policies commonly requires to know the actual drivers of behavior. This paper provides one possible answer to that question and stresses the importance of de-veloping mechanisms which allow to distinguish between different causes of probability distortions. Apparent violations of expected utility theory, and especially violations of the assumptions that preferences are linear in probabilities (i.e. the independence axiom), may be due to environmental constraints rather than to a failure of the standard prefer-ence model. This is an important finding. Preferprefer-ences, as a central primitive of economic behavior, should be stable across different choice domains and over time.8 Evidence sug-gest that they are not (e.g. Deck et al. (2009) and Zeisberger et al. (2010)). This issue may be resolved by taking into account exogenous factors affecting choice behavior. In fact, our model shows that it is possible to accommodate the most important behavioral patterns of choice under risk without discarding or needing to generalize expected utility.
While both, our model and models incorporating probability weighting capture similar properties of risk taking behavior, there are many situations for which their predictions differ in fundamental ways. For example, our model predicts that risk taking behavior depends on both, stake size and choice domain. Decision makers are predicted to depart less strongly from linear probability weighting as stakes grow larger or the commodity of interest in the respective choice domain becomes less scarce, ceteris paribus. Facilitating market access may be one possible policy intervention encouraging economic agents to act in a way closer to their true preferences. In contrast, RDU models do neither make such predictions, nor do they point at solutions for such problems. They only capture revealed preferences and not the mechanism generating behavior. RDU parameter estimates stem-ming from one study should therefore only be used with care when making predictions or calibrating models for different choice domains, different subjects or different points in time.
The remainder of this paper is organized as follows. Section C.2 reviews the related literature. Section C.3 derives a simple model of choice under risk and fleshes out its basic properties leading to departures from standard expected utility. Section C.4 presents the
The remainder of this paper is organized as follows. Section C.2 reviews the related literature. Section C.3 derives a simple model of choice under risk and fleshes out its basic properties leading to departures from standard expected utility. Section C.4 presents the